An organism develops and functions according to the genes contained in its cells. In nature, genes from one species do not mix with genes from a different species, because organisms from different species cannot reproduce. The process of genetic modification overcomes this natural barrier, because it allows scientists to transfer genetic traits from one organism to a completely unrelated organism. For example, certain varieties of corn (Zea mays) have been genetically modified for pest resistance through the addition of a gene taken from a bacterium known as Bacillus thuringiensis.
Hybrid plant varieties are the result of special breeding techniques, but these techniques do not involve the transference of genes from one species to another. A hybrid is simply the first-generation offspring of carefully controlled cross-pollination between two stable varieties. Breeders can use this type of cross-pollination to produce hybrid plants that are more vigorous, productive, uniform or stress-tolerant than the parent varieties. Nevertheless, some gardeners avoid hybrids because they do not breed true -- in other words, a hybrid's characteristics are not genetically stable, so seed saved from a hybrid variety will not reliably produce plants that resemble the parent plant.
Seedless watermelons are the result of a complex process that involves hybridization as well as other advanced breeding techniques. Genes are manipulated to produce plants that cannot properly reproduce because they have three sets of chromosomes. This manipulation is certainly unnatural, but seedless watermelons are not genetically modified because they do not contain genes from unrelated organisms. In regards to saving seed, seedless watermelons are even worse than hybrids -- they do not breed true, but more importantly, they do not form mature seeds at all, so there is nothing to save. The small white seeds in seedless watermelons are empty seeds coats that will not germinate.
Home gardeners rarely grow seedless watermelons because they require specialized cultural practices and because the seed is expensive. Seedless varieties germinate poorly, so you need to carefully control the temperature (85 degrees Fahrenheit) of the germination medium. You also need to avoid any form of temperature stress during the seedling stage. Perhaps the most important cultural technique is interplanting a pollinator variety among the seedless varieties. The purpose of these pollinator plants is simply to provide pollen for the flowers of the seedless plants, which cannot produce their own pollen.